On uniqueness of submaximally symmetric parabolic geometries

Abstract

Among (regular, normal) parabolic geometries of type (<i>G,P</i>), there is a locally unique maximally symmetric structure and it has symmetry dimension dim(<i>G</i>). The symmetry gap problem concerns the determination of the next realizable (submaximal) symmetry dimension. When <i>G</i> is a complex or split-real simple Lie group of rank at least three or when (<i>G,P</i>) = (<i>G₂</i>, <i>P</i>₂), we establish a local uniqueness result for submaximally symmetric structures of type (<i>G,P</i>).